Non-mopar tech question, sorry...

Sounds like a vacuum issue. Look under the hood over by the heater control valve is. Look at the vacuum line going into the valve. They are bad about rotting and even sometimes falling onto the exhaust manifold.

Also look at every other vacuum line you can find. Something is either rotted and leaking or disconnected and leaking, causing the blend doors to move from lost vacuum when you accelerate.

I don't think my truck has a heater control valve. My heater hoses come out of the firewall and go straight to the engine block. I could be wrong, though. I'm not a mechanic, and I surely don't know everything. I'll still have a look at the other lines I can find and see if anything is messed up. Thanks for the suggestion.

Check the input and output hoses are both getting hot ?

X2
An easy way to tell if you have a flow problem.
If one hose is good and warm and the other is cold, then disconnect them both and flush the heater core in both directions.

(chk the valve as RRR mentioned at the same time, as I'll bet these two checks will give you your answer)

I did feel the hoses after I swapped the thermostat and let the truck warm up. The inboard hose was really hot and the outboard hose was lukewarm to the touch, at best. Personally, I think I'd rather just swap the heater core with a new one. They're only $25+tax, and it takes about 30 minutes with my truck. Undo the clamps on the heater hoses and pull them off, pull clips out from under the dash and remove a trim panel, pull four bolts off the cover to the heater core, and then just pull it out. To put the new one in, just rinse and repeat. I'll let you all know how it turns out.

Thanks everyone!